Trigonometry of the quantum state space, geometric phases and relative phases

Abstract

A complete set of invariants for three states in the quantum space of states is obtained together with a complete set of relationships linking them. This is done in a way that preserves the self-duality of and leads to a clear geometric description of invariants (distances, lateral phases; Hermitian angles, angular phases; and two purely triangular phases). Some of these invariants appear here for the first time. Symplectic area (and hence the triangle geometric phase) is proportional to a 'mixed phase excess', thus extending to the relation 'area-angular excess' in the real sphere. The new triangle lateral phases provide a description, intrinsic to , of relative phases in a superposition. This approach also provides closed expressions for the triangle holonomy associated with the usual Fubini–Study metric in , as well as many other expressions for similar 'loop' operators along the triangle, including closed and exact expressions for the triangle Aharonov–Anandan geometric phase.